In this work, certain quasilinear elliptic boundary value problems are investigated. Homogeneous Dirichlet boundary condition is always considered. In the first result, assuming that the multivalued monotone nonlinearity \(\beta\) satisfies \(\operatorname {dom}\beta = R\) and the existence of an upper and a lower solution, the existence of a solution between them is proved. For the case of \(\operatorname {dom}\beta \neq \mathbb{R}\), the theory of monotone operators is used to show the existence of a solution in a restricted situation. Finally, in the case of multivalued nonmonotone nonlinearity, an eigenvalue problem is considered. Using critical point theory for nonsmooth locally Lipschitz functionals, the existence of (at least) two nontrivial solutions is proved.